Two simple harmonic motions are represented by the equations $x_{1}=5 \sin \left(2 \pi t+\frac{\pi}{4}\right)$ and $x_{2}=5 \sqrt{2}(\sin 2 \pi t+\cos 2 \pi t)$. The ratio of the amplitude of $x_{1}$ and $x_{2}$ is

Two simple harmonic motions are represented by the equations $x_{1} = 5 \sin(2 \pi t + \frac{\pi}{4})$ and $x_{2} = 5 \sqrt{2}(\sin 2 \pi t + \cos 2 \pi t)$. The amplitude of the second motion is ....... times the amplitude of the first motion.

$A$ vibratory motion is represented by $x = 2A \cos \omega t + A \cos \left( \omega t + \frac{\pi}{2} \right) + A \cos ( \omega t + \pi ) + \frac{A}{2} \cos \left( \omega t + \frac{3\pi}{2} \right)$. The resultant amplitude of the motion is

The displacement of a particle in a periodic motion is given by $y = 4 \cos^{2}\left(\frac{t}{2}\right) \sin(1000 t)$. This displacement may be considered as the result of the superposition of $n$ independent harmonic oscillations. Here $n$ is

The amplitude of the vibrating particle due to the superposition of two $SHMs$,$y_1 = \sin \left( \omega t + \frac{\pi}{3} \right)$ and $y_2 = \sin \omega t$ is:

$A$ point mass is subjected to two simultaneous sinusoidal displacements in $x$-direction,$x_1(t) = A \sin \omega t$ and $x_2(t) = A \sin \left(\omega t + \frac{2 \pi}{3}\right)$. Adding a third sinusoidal displacement $x_3(t) = B \sin (\omega t + \phi)$ brings the mass to a complete rest. The values of $B$ and $\phi$ are